Can the Grothendieck game self-test continuous groups of quantum measurements?

Continuous questions force a probabilistic self-testing guarantee

Standard self-tests use games with finitely many questions, so every designated measurement is asked with positive probability and can be pinned down individually. The Grothendieck game instead samples question pairs (x, y) of vectors on the unit sphere S^(n-1) with density |⟨x, y⟩|/z_n relative to the uniform product measure, where z_n is the normalization constant. Under a continuous distribution, any single question has probability zero, so an optimal strategy can be altered arbitrarily on any fixed finite set of questions without changing its value.

That is why the paper argues no fixed finite set of generators can be certified, and why even certifying finitely many generators would not constrain the measurements at the remaining questions. The guarantee the authors aim for is probabilistic: a near-optimal strategy should approximate an ideal implementation on average, or with high probability over the sampled questions, under one pair of local isometries. The paper states that self-testing a continuous family of measurements from a near-optimal strategy for a single nonlocal game had not previously been studied; the closest related work certifies arbitrary real projective measurements through an iterative procedure that adds one measurement at a time.

How the Grothendieck game turns sphere directions into binary answers

The Grothendieck game G_n is an XOR game: the referee sends question x to Alice and question y to Bob, and the players win if their answers a, b in {±1} satisfy ab = sgn⟨x, y⟩. In words, their answers must agree when the inner product is positive and differ when it is negative. The bias of a strategy with winning probability p is 2p − 1, and the optimal classical bias is β_c(G_n) = z_n while the optimal quantum bias is β_q(G_n) = (1/n) z_n.

The game is a natural candidate for certifying a continuous family because the sharpness proof of the little Grothendieck inequality makes G_n universal: its optimal strategy yields optimal quantum strategies for any finite positive semidefinite XOR game. The paper notes that G_n had previously been studied mainly as a game requiring a large amount of entanglement when played optimally, and that its self-testing properties had not been studied.

Majorana observables form the ideal continuous strategy

For even n = 2m ≥ 2, the Majorana operators γ_1, …, γ_n on C^(2^m) satisfy γ_i γ_j + γ_j γ_i = 2δ_ij I. They generate the unique irreducible 2^m-dimensional representation of the finite Clifford group C_n, and any orthonormal basis of R^n gives operators satisfying the same relations. Defining γ(x) = Σ_i x_i γ_i for x on the unit sphere gives γ(x)* = γ(x), γ(x)^2 = I, and {γ(x), γ(y)} = 2⟨x, y⟩ I, so every γ(x) is a binary observable.

An optimal quantum strategy for G_n uses a maximally entangled state, with Alice measuring γ(x) and Bob measuring γ(y) on their respective local Hilbert spaces. Because varying the orthonormal basis yields a continuous family of equivalent representations of C_n, the object to certify is not a finite list of operators but this whole family. The paper states that self-tests for n Majorana operators satisfying the anticommutation relations were already known, but certifying the entire continuous family simultaneously had not been achieved.

Near-optimal bias certifies the whole Majorana family

Theorem 1.1 (Theorem 5.8 in the paper) states that for n = 2m, if a finite-dimensional strategy for G_n with state |ψ⟩ and observables A(x), B(y) achieves bias at least (1 − ε) β_q(G_n), then there exist local isometries Φ_A, Φ_B and an auxiliary state |aux⟩ such that the state is close to the 2^m-dimensional maximally entangled state up to O(n^2 ε), and the expected deviations of A(x) from γ(x) and of B(y) from γ(y) over the question distribution are each bounded by O(n ε).

The expectation over questions is not a technical convenience but a necessity: because changing measurements at any finite set of questions leaves the bias unchanged, no fixed finite set can be certified. To recover guarantees for fixed operators, Corollary 5.10 deduces from Theorem 1.1 that in a Haar-random orthonormal basis Q = (q_1, …, q_n) of R^n, all 2n observables A(q_i) and B(q_i) are simultaneously certified with high probability. The robustness bounds are therefore stated in two different currencies — an average over the continuous question distribution, and a high-probability statement over a randomly drawn basis — and the paper treats them as serving different certification needs.

Products of Majorana observables certify the pin group

The operators γ(x) can be multiplied to generate an irreducible representation of the pin group Pin(n). Writing self-adjoint unitaries c(x) for x on the sphere subject to c(x)c(y) + c(y)c(x) = 2⟨x, y⟩, the spin representation Γ maps a product c(x_1)⋯c(x_k) to γ(x_1)⋯γ(x_k). Any element of Pin(n) can be written as a signed product of at most n generators, but that decomposition is not unique, so it is not a priori clear how Alice's and Bob's operators A(x) and B(y) extend to an approximate representation of the whole group.

The paper resolves this by bringing in rotations R in SO(n) and their associated automorphisms of the pin group, obtaining unitaries A_R(g) and B_R(g) for g in Pin(n) and R in SO(n) that Alice and Bob can implement given unitary access to their observables. Theorem 1.2 (Theorem 5.11) then gives, for the same isometries and state as Theorem 1.1, expected deviations bounded by O(n^3 ε) over uniformly random R in SO(n) and g in Pin(n). The price of lifting the guarantee from generators to the full group is visible in the exponent: the error grows from O(n ε) for the Majorana observables to O(n^3 ε) for the spin representation.

Rounding approximate representations with an efficient isometry

A key proof step is rounding an approximate representation of the Clifford group C_n to an exact one. The paper presents a variant of the Gowers–Hatami stability theorem that applies to finite and compact continuous groups alike, allows targeting a specific irreducible representation, and clarifies what is needed for the isometry to be implemented efficiently. For any function f from a group G to unitaries on a Hilbert space H, the map sending |ψ⟩ to the function g ↦ f(g)|ψ⟩ is an isometry, and the deviation from an exact representation is exactly the average over G of ‖f(g)f(h) − f(gh)‖^2 in the state-weighted norm.

Targeting a chosen irreducible representation λ is achieved by projecting onto the λ-isotypical subspace, which the Fourier transform of G identifies with End(H_λ) ⊗ H. The resulting isometry V_λ can be implemented by preparing a uniform superposition over the group, applying the controlled approximate representation, and then applying the group Fourier transform. The paper states that V_λ is efficient whenever the uniform superposition, the controlled approximate representation, and the Fourier transform can be implemented efficiently — a conditional statement about implementation cost, not a claim that every group in this framework admits an efficient circuit.

Classical and vector strategies are rigid too

The paper also establishes rigidity for the simpler strategy classes of G_n. An optimal deterministic strategy is given by measurable functions a(x) = sgn⟨c, x⟩ and b(y) = sgn⟨c, y⟩ for some c on the sphere, and Theorem 3.1 shows that any near-optimal deterministic strategy is close to such a strategy. For vector strategies, which assign real unit vectors a(x) and b(y) to the questions with correlation ⟨a(x), b(y)⟩, Tsirelson's theorem makes the optimal vector and quantum biases coincide, and Theorem 4.8 proves that every near-optimal vector strategy is close to the strategy a(x) = x, b(y) = y.

These results matter for reading the quantum theorems because they show the rigidity of G_n is not an artifact of one proof technique: the same game pins down its classical, vector, and quantum near-optimal strategies, each in its own language. They also mark the boundary of what the paper claims. The self-testing statements are about finite-dimensional strategies for G_n, and the robustness bounds scale with n, so the certified closeness degrades as the dimension of the sphere grows.

What the guarantee does and does not cover

The paper's central claim is that a single nonlocal game, G_n, robustly self-tests a continuous family of measurements: the Majorana observables γ(x) for all x on the sphere, and through products of those observables the spin representation of the continuous pin group. The certification is probabilistic by construction, since a continuous question distribution assigns probability zero to any single question, and the paper is explicit that this is why the guarantee is stated as an average over questions or as a high-probability statement over a Haar-random basis.

The most decision-relevant comparison in the paper is between the two robustness scales. Certifying the Majorana observables costs O(n ε) in the expected deviation, while certifying the full pin group spin representation costs O(n^3 ε) over uniformly random rotations and group elements. A reader weighing whether the Grothendieck game is the right self-test for a given certification task should therefore look first at which object needs to be certified — the generators alone or the group they generate — because that choice, not the game itself, determines the error scaling the paper actually proves.

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